The cohomology of $\text{Gr}_3(\mathbb{R}^{6,3})$

For details of this computation, see Section 6 of this paper .

Poincare polynomial: $$x^{9} y^{5} + x^{8} y^{4} + 2 x^{7} y^{4} + x^{6} y^{4} + 2 x^{6} y^{3} + 3 x^{5} y^{3} + x^{4} y^{3} + 2 x^{4} y^{2} + x^{3} y^{3} + 2 x^{3} y^{2} + x^{2} y^{2} + x^{2} y + x y + 1$$ Generator count grid:
1 1 2 2 1 2 2 3 3 3 1 2 2 1 2 2 1 1 1 1
Explicitly, as a free module over the ground ring $\mathbb{M}_2$: $$H^{\ast,\ast}(\text{Gr}_3(\mathbb{R}^{6,3}))=\mathbb{M}_2\oplus\Sigma^{1,1}\mathbb{M}_2\oplus\Sigma^{2,1}\mathbb{M}_2\oplus\Sigma^{2,2}\mathbb{M}_2\oplus\Sigma^{3,2}\mathbb{M}_2\oplus\Sigma^{3,3}\mathbb{M}_2\oplus\Sigma^{4,2}\mathbb{M}_2\oplus\Sigma^{4,3}\mathbb{M}_2\oplus\Sigma^{5,3}\mathbb{M}_2\oplus\Sigma^{6,3}\mathbb{M}_2\oplus\Sigma^{6,4}\mathbb{M}_2\oplus\Sigma^{7,4}\mathbb{M}_2\oplus\Sigma^{8,4}\mathbb{M}_2\oplus\Sigma^{9,5}\mathbb{M}_2.$$